Light and Matter Tim Freegarde School of Physics & Astronomy University of Southampton Controlling light with matter

2 Optical polarization for any wavevector, there are two field components light is a transverse wave: perpendicular to any wave may be written as a superposition of the two polarizations

3 The Fresnel equation return to derivation of electromagnetic wave equation consider oscillatory waves of definite polarization a ω apply vector identity twice and simplify

4 The Fresnel equation return to derivation of electromagnetic wave equation consider oscillatory waves of definite polarization a ω apply vector identity twice and simplify

5 The Fresnel equation for isotropic media electromagnetic waves are transverse the Poynting vector is parallel to the wavevector

6 Characterizing the optical polarization wavevector insufficient to define electromagnetic wave we must additionally define the polarization vector e.g. linear polarization at angle

7 Jones vectors normalized polarization vector is known as the Jones vector real field corresponds to superposition of exponential form and complex conjugate defines polarization state of any wave of given and

8 Categories of optical polarization linear (plane) polarization coefficients differ only by real factor circular polarization coefficients differ only by factor elliptical polarization all other cases

9 Categories of optical polarization linear (plane) polarization coefficients differ only by real factor circular polarization coefficients differ only by factor elliptical polarization all other cases

10 Polarization notation circular polarization right- or left-handed rotation when looking towards source linear (plane) polarization parallel or perpendicular to plane of incidence RCP plane of incidence perpendicular parallel traces out right- or left-handed thread plane of incidence contains wavevector and normal to surface

11 Categories of optical polarization complex electric field given by real electric field corresponds to superposition with complex conjugate for monochromatic fields, Jones vector is constant

12 Polarization of time-varying fields complex polychromatic electric field given by beating between frequencies causes field to vary with time even stabilized lasers have linewidth in the MHz range Jones vector may therefore vary on a microsecond timescale – or faster

13 Stokes parameters with polychromatic light, the Jones vector varies we therefore describe polarization through averages and correlations: STOKES PARAMETERS are the instantaneous field components is their relative phase

14 Stokes parameters with polychromatic light, the Jones vector varies we therefore describe polarization through averages and correlations: STOKES PARAMETERS are the instantaneous field components is their relative phase

15 Stokes parameters total intensity, I 1 related to horizontally polarized component, I 2 … component polarized at +45º to horizontal, I 3 … right circularly polarized component, I 4

16 Unpolarized (randomly polarized) light average horizontal component = average vertical component average +45º component = average -45º component average RCP component = average LCP component … = half total intensity orthogonal polarizations are uncorrelated

17 Degree of polarization for partially polarized light, the quantity represents the degree of polarization, where unpolarized (randomly polarized) completely polarized

18 Completely polarized light constant Jones vector Stokes parameters given by when simply defining the polarization state, it is common to drop the intensity factor I 1

19 The Poincaré sphere (a)right circularly polarized plot the Stokes vector (b)left circularly polarized (c)horizontally polarized (d)vertically polarized (e)polarized at +45º (f)elliptically polarized (a) (b) (c) (d) (e) (f) (g)unpolarized right elliptically polarized (g) left elliptically polarized [0, 0, 1] [0, 0,-1] [1, 0, 0] [-1, 0, 0] [0, 1, 0] [δ 1,δ 2,δ 3 ]/δ 0 [0, 0, 0]

20 Polarizers many optical elements restrict or modify the polarization state of light polarization-dependent transmission/reflection polarization-dependent refractive index sheet polarizers (Polaroid) Nicol, Wollaston prisms etc polarizers, polarizing filters, analyzers waveplates, retarders four categories of physical phenomenapolarization-sensitive absorption (dichroism) polarization-sensitive dispersion (birefringence, optical activity) reflection at interfaces scattering

21 Polarizers each mechanism may discriminate between either linear or circular polarizations mechanisms depend upon an asymmetry in the device or medium plane of incidence perpendicular parallel

22 Linear polarization upon reflection for normal incidence, no distinction between horizontal and vertical polarizations if wavevector makes angle with interface normal, s- and p-polarizations affected differently we consider here the reflection of p-polarized light; s-polarized beams may be treated similarly we resolve the electric field into components parallel and normal to the interface all magnetic field components are parallel to the interface

23 Linear polarization upon reflection for normal incidence, no distinction between horizontal and vertical polarizations if wavevector makes angle with interface normal, s- and p-polarizations affected differently we consider here the reflection of p-polarized light; s-polarized beams may be treated similarly we resolve the electric field into components parallel and normal to the interface all magnetic field components are parallel to the interface combine forward and reflected waves to give total fields for each region apply continuity conditions for separate components hence derive fractional transmission and reflection

24 Fresnel equations p-polarization s-polarization combine forward and reflected waves to give total fields for each region apply continuity conditions for separate components hence derive fractional transmission and reflection

25 Linear dichroism conductivity of wire grid depends upon field polarization electric fields perpendicular to the wires are transmitted WIRE GRID POLARIZER fields parallel to the wires are absorbed

26 Linear dichroism crystals may similarly show absorption which depends upon linear polarization absorption also depends upon wavelength polarization therefore determines crystal colour TOURMALINE pleochroism, dichroism, trichroism

27 Circular dichroism absorption may also depend upon circular polarization the scarab beetle has polarization- sensitive vision, which it uses for navigation the beetle’s own colour depends upon the circular polarization SCARAB BEETLELEFT CIRCULAR POLARIZED LIGHT RIGHT CIRCULAR POLARIZED LIGHT

28 Polarization in nature the European cuttlefish also has polarization-sensitive vision … and can change its colour and polarization! MAN’S VIEWCUTTLEFISH VIEW (red = horizontal polarization) CUTTLEFISH (sepia officinalis)

29 Birefringence asymmetry in crystal structure causes polarization dependent refractive index opposite polarizations follow different paths through crystal birefringence, double refraction

º Linear polarizers (analyzers) e -ray o -ray e -ray o -ray s -ray p -ray birefringence results in different angles of refraction and total internal reflection many different designs, offering different geometries and acceptance angles a similar function results from multiple reflection

31 Waveplates (retarders) WAVEPLATE at normal incidence, a birefringent material retards one polarization relative to the other linearly polarized light becomes elliptically polarized

32 Compensators adjust fixed variable a variable waveplate uses two wedges to provide a variable thickness of birefringent crystal a further crystal, oriented with the fast and slow axes interchanged, allows the retardation to be adjusted around zero SOLEIL COMPENSATOR with a single, fixed first section, this is a ‘single order’ (or ‘zero order’) waveplate for small constant retardation

33 Optical activity (circular birefringence) l-limonene (orange) r-limonene (lemon) CH 2 CH 3 H CH 2 CH 3 H CHIRAL MOLECULES optical activity is birefringence for circular polarizations an asymmetry between right and left allows opposing circular polarizations to have differing refractive indices optical activity rotates the polarization plane of linearly polarized light may be observed in vapours, liquids and solids

34 Jones vector calculus if the polarization state may be represented by a Jones vector then the action of an optical element may be described by a matrix JONES MATRIX

35 Jones vector calculus if the polarization state may be represented by a Jones vector then the action of an optical element may be described by a matrix JONES MATRIX transmission by horizontal polarizer retardation by waveplate projection onto rotated axes

36 Müller calculus MŰLLER MATRIX field averages and correlations following optical element depend linearly upon parameters describing incident beam Müller matrix elements may be written in terms of Jones matrix elements, e.g.

37 Müller calculus the actions of optical materials can be represented by geometrical transformations of the Stokes vector in the Poincaré sphere optical activity: rotation about a vertical axis right elliptically polarized left elliptically polarized

38 Müller calculus the actions of optical materials can be represented by geometrical transformations of the Stokes vector in the Poincaré sphere optical activity: rotation about a vertical axis birefringence: rotation about a horizontal axis left elliptically polarized right elliptically polarized